42 research outputs found
On the axioms for adhesive and quasiadhesive categories
A category is adhesive if it has all pullbacks, all pushouts along
monomorphisms, and all exactness conditions between pullbacks and pushouts
along monomorphisms which hold in a topos. This condition can be modified by
considering only pushouts along regular monomorphisms, or by asking only for
the exactness conditions which hold in a quasitopos. We prove four
characterization theorems dealing with adhesive categories and their variants.Comment: 20 pages; v2 final version, contains more details in some proof
An embedding theorem for adhesive categories
Adhesive categories are categories which have pushouts with one leg a
monomorphism, all pullbacks, and certain exactness conditions relating these
pushouts and pullbacks. We give a new proof of the fact that every topos is
adhesive. We also prove a converse: every small adhesive category has a fully
faithful functor in a topos, with the functor preserving the all the structure.
Combining these two results, we see that the exactness conditions in the
definition of adhesive category are exactly the relationship between pushouts
along monomorphisms and pullbacks which hold in any topos.Comment: 8 page
The univalence axiom for elegant Reedy presheaves
We show that Voevodsky's univalence axiom for intensional type theory is
valid in categories of simplicial presheaves on elegant Reedy categories. In
addition to diagrams on inverse categories, as considered in previous work of
the author, this includes bisimplicial sets and -spaces. This has
potential applications to the study of homotopical models for higher
categories.Comment: 25 pages; v2: final version, to appear in HH
Characterizing Van Kampen Squares via Descent Data
Categories in which cocones satisfy certain exactness conditions w.r.t.
pullbacks are subject to current research activities in theoretical computer
science. Usually, exactness is expressed in terms of properties of the pullback
functor associated with the cocone. Even in the case of non-exactness,
researchers in model semantics and rewriting theory inquire an elementary
characterization of the image of this functor. In this paper we will
investigate this question in the special case where the cocone is a cospan,
i.e. part of a Van Kampen square. The use of Descent Data as the dominant
categorical tool yields two main results: A simple condition which
characterizes the reachable part of the above mentioned functor in terms of
liftings of involved equivalence relations and (as a consequence) a necessary
and sufficient condition for a pushout to be a Van Kampen square formulated in
a purely algebraic manner.Comment: In Proceedings ACCAT 2012, arXiv:1208.430
Graph Rewriting and Relabeling with PBPO+
We extend the powerful Pullback-Pushout (PBPO) approach for graph rewriting
with strong matching. Our approach, called \pbpostrong, exerts more control
over the embedding of the pattern in the host graph, which is important for a
large class of graph rewrite systems. In addition, we show that \pbpostrong is
well-suited for rewriting labeled graphs and certain classes of attributed
graphs. For this purpose, we employ a lattice structure on the label set and
use order-preserving graph morphisms. We argue that our approach is simpler and
more general than related relabeling approaches in the literature.Comment: 20 pages, accepted to the International Conference on Graph
Transformation 2021 (ICGT 2021
A Unifying Theory for Graph Transformation
The field of graph transformation studies the rule-based transformation of graphs. An important branch is the algebraic graph transformation tradition, in which approaches are defined and studied using the language of category theory. Most algebraic graph transformation approaches (such as DPO, SPO, SqPO, and AGREE) are opinionated about the local contexts that are allowed around matches for rules, and about how replacement in context should work exactly. The approaches also differ considerably in their underlying formal theories and their general expressiveness (e.g., not all frameworks allow duplication). This dissertation proposes an expressive algebraic graph transformation approach, called PBPO+, which is an adaptation of PBPO by Corradini et al. The central contribution is a proof that PBPO+ subsumes (under mild restrictions) DPO, SqPO, AGREE, and PBPO in the important categorical setting of quasitoposes. This result allows for a more unified study of graph transformation metatheory, methods, and tools. A concrete example of this is found in the second major contribution of this dissertation: a graph transformation termination method for PBPO+, based on decreasing interpretations, and defined for general categories. By applying the proposed encodings into PBPO+, this method can also be applied for DPO, SqPO, AGREE, and PBPO
Computational category-theoretic rewriting
We demonstrate how category theory provides specifications that can
efficiently be implemented via imperative algorithms and apply this to the
field of graph rewriting. By examples, we show how this paradigm of software
development makes it easy to quickly write correct and performant code. We
provide a modern implementation of graph rewriting techniques at the level of
abstraction of finitely-presented C-sets and clarify the connections between
C-sets and the typed graphs supported in existing rewriting software. We
emphasize that our open-source library is extensible: by taking new categorical
constructions (such as slice categories, structured cospans, and distributed
graphs) and relating their limits and colimits to those of their underlying
categories, users inherit efficient algorithms for pushout complements and
(final) pullback complements. This allows one to perform double-, single-, and
sesqui-pushout rewriting over a broad class of data structures